{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "41912"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import json\n",
    "\n",
    "datasets = [\"amc\", \"omni_math\", \"aime\", \"still\", \"math\"]\n",
    "\n",
    "data = []\n",
    "for dataset in datasets:\n",
    "    with open(f\"../train/{dataset}.json\") as f:\n",
    "        data.extend(json.load(f))\n",
    "# with open(f'../train/omni_math.json', 'r') as f:\n",
    "#     data.extend(json.load(f))\n",
    "len(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\\lceil \\log_2 n \\rceil\n",
      "\\frac{1}{2} \\max A + 3\n",
      "\\frac{4}{\\sqrt{3}} \\sin^2 80^\\circ\n",
      "K \\text{ is the intersection of } AB' \\text{ and } BA', \\text{ and } c \\text{ is a constant}\n",
      "f(x, y) = C \\text{ for some constant } C \\in [0, 1]\n",
      "(2, 6), (2^k - 1, 2), (1, n) \\text{ for any } n \\ge 1\n",
      "4122 \\text{ (minimum)}, 6049 \\text{ (maximum)}\n",
      "\\text{All positive integers } n \\text{ relatively prime to } 101!\n",
      "f(x) = x \\text{ or } f(x) = \\begin{cases} n & \\text{if } x = 1 \\\\ 1 & \\text{if } x > 1 \\end{cases} \\text{ or } f(x) = \\begin{cases} n & \\text{if } x = 1 \\\\ 1 & \\text{if } x > 1 \\text{ is odd} \\\\ 2 & \\text{if } x \\text{ is even} \\end{cases} \\text{ for any } n \\text{ odd}\n",
      "P(x,y,z) = x^2 + y^2 + z^2 + 2xyz\n",
      "\\text{the incenter, circumcenter, and orthocenter of } \\triangle ABC\n",
      "\\text{linear functions and downward-facing parabolas}\n",
      "n \\text{ works if and only if the set of prime divisors of } n \\text{ is not the set of the first } k \\text{ primes for some } k\n",
      "\\sum_{i=1}^n \\min \\{ n + 1 - i, 2i-1 \\}\n",
      "(2, 2) \\text{ and } ((i+1)^2 + 1, (i+2)^2 + 1) \\text{ for all } i \\in \\mathbb{N}\n",
      "2017^{98}\n",
      "W(a,b) = k \\text{ for all distinct } a, b \\text{ and } k = 1 \\text{ or } k = -1.\n",
      "f(x) = 0 \\ \\ \\forall x \\in \\mathbb{R}\n",
      "\\text{No such } n \\text{ exists}\n",
      "\\text{not a rational number}\n",
      "\\begin{cases}\n",
      "n^2 - n + 1 & \\text{if } n \\text{ is odd}, \\\\\n",
      "n^2 & \\text{if } n \\text{ is even}.\n",
      "\\end{cases}\n",
      "f(m) = km \\text{ for any positive integer constant } k.\n",
      "n + v_p(n!)\n",
      "f(x) = ax \\text{ for } a \\geq 0\n",
      "f(x) = 1 \\text{ for all } x \\in \\mathbb{N}\n",
      "\\prod_{i=1}^k p_i^{p_i} \\text{ for part (a)}\n",
      "k = x(a+1) - 1 \\text{ and } p = 2^{a+1} - 1 \\text{ for } a, x \\in \\mathbb{N}\n",
      "\\text{All odd positive integers}\n",
      "\\min(m, n)^{\\frac{1}{r} - \\frac{1}{s}}\n",
      "p^a q^b r^c\n",
      "f(x,y) = a + \\min(x,y) \\quad \\text{or} \\quad f(x,y) = a + \\max(x,y) \\quad \\text{for any } a \\in \\mathbb{R}.\n",
      "f(x) = g(x) = 0 \\text{ for all } x \\in \\mathbb{R} \\text{ or } f(x) = g(x) \\text{ with } f(0) = 0\n",
      "t = \\max \\{ 2q, q - 1 + 2M \\}\n",
      "n \\text{ must be odd}\n",
      "f(n) = cn\n",
      "2^k \\cdot m \\text{ where } k = 1 \\text{ and } m \\text{ is an odd integer}\n",
      "\\text{No, there does not exist such a prime } p.\n",
      "\\text{All composite integers } s \\geq 4\n",
      "\\[\n",
      "\\begin{cases} \n",
      "n^2 + 4 & \\text{if } n \\text{ is even} \\\\\n",
      "n^2 + 3 & \\text{if } n \\text{ is odd}\n",
      "\\end{cases}\n",
      "\\]\n",
      "\\text{circles passing through fixed points}\n",
      "\\text{Yes}\n",
      "a = -1 - pq \\text{ or } a = 1 + (-1)^n pq\n",
      "\\[\n",
      "a = 0, \\quad b = 2, \\quad c = 2, \\quad d = 1, \\quad e = 0, \\quad f = 2\n",
      "\\]\n",
      "(a_1, a_2, \\ldots, a_n) \\text{ where } a_1 = k \\cdot 2^n + 1 \\text{ and } a_2, \\ldots, a_n \\text{ are odd integers such that } 1 < a_1 \\le a_2 \\le \\cdots \\le a_n\n",
      "\\text{All } k \\text{ such that } v_p(k) \\geq 2018 \\text{ for some prime } p \\text{ or } v_q(k) \\geq 1009 \\text{ and } v_r(k) \\geq 2 \\text{ for some distinct primes } q \\text{ and } r.\n",
      "The locus of the point of intersection of lines \\(AX\\) and \\(BY\\) is a circle with the equation:\n",
      "\\[\n",
      "x^2 + \\left(y - \\frac{1}{b}\\right)^2 = \\frac{a^2}{b^2}.\n",
      "\\]\n",
      "\\{ n \\in \\mathbb{Z}^+ \\mid n \\geq 4 \\text{ and } n \\text{ is even} \\}\n",
      "\\[\n",
      "(a, b, c) \\rightarrow \\boxed{(n\\sqrt{2}, n, n)}\n",
      "\\]\n",
      "The assertion is false; \\( n = 25 \\) is a counter-example.\n",
      "\\[ f(x) = \\frac{k}{1+x} + \\frac{1-k}{3} \\quad \\left(-\\frac{1}{2} \\le k \\le 1\\right) \\]\n",
      "\\[\n",
      "(a, b, c) = (k, k+1, k+2) \\quad \\text{and all cyclic permutations, with } k \\in \\mathbb{Z}\n",
      "\\]\n",
      "All rational numbers between 0 and 1 inclusive will eventually yield some \\( x_k = 0 \\).\n",
      "The roots of the system of simultaneous equations are \\(x = 1\\), \\(y = 1\\), and \\(z = 1\\).\n",
      "\\[ n = 2^{a+1} + 2^b, \\quad a, b \\ge 0 \\]\n",
      "\n",
      "Alternatively, this condition can be expressed as either \\( n = 2^k, \\, k \\ge 2 \\) or \\( n \\) is the sum of two distinct powers of 2.\n",
      "\\[\n",
      "\\frac{\\cos 1^\\circ}{\\sin^2 1^\\circ}\n",
      "\\]\n",
      "\\[ k = 2^{27} - 2^{11} \\]\n",
      "The only positive integers \\( m \\) that will work are numbers in the form of \\( xy^2 \\), other than \\( 1 \\), for integers \\( x \\) and \\( y \\) (where \\( x \\) and \\( y \\) can be equal), i.e., \\( 4, 8, 9, 12, 16, 18, 20, 24, 25, \\dots \\).\n",
      "The smallest possible value of \\(a_{10}\\) is \\(248\\).\n",
      "The only solutions are \\((2, 3, 3)\\) and its permutations.\n",
      "All pairs \\((m, n)\\) such that \\(m\\) and \\(n\\) are odd and \\(m+n\\) is a positive power of 2.\n",
      "The problem does not have a provided solution, so the final answer cannot be extracted.\n",
      "-512 \\leq (a+b)(b+c)(c+d)(d+e)(e+a) \\leq 288\n",
      "\\[\n",
      "\\prod_{i=1}^{1005}(4i-1) = 3 \\times 7 \\times \\ldots \\times 4019\n",
      "\\]\n",
      "The integers \\( n \\geq 3 \\) that satisfy the given property are \\( n = 3 \\) and \\( n = 4 \\).\n",
      "The maximum number of towns is \\(4\\).\n",
      "\\[ 0 < \\angle APC + \\angle BPD < \\pi \\]\n",
      "The possible values of \\( f(1000) \\) are all even numbers.\n",
      "\\[\n",
      "\\text{(i)} \\quad \\text{For all positive integers } n.\n",
      "\\]\n",
      "\\[\n",
      "\\text{(ii)} \\quad \\text{No, there is no such infinite set } S.\n",
      "\\]\n",
      "\\[ P(x) = \\frac{x\\left(x^{4k+2}+1\\right)}{x^{2}+1} \\quad \\text{or} \\quad P(x) = \\frac{x\\left(1-x^{4k}\\right)}{x^{2}+1} \\]\n",
      "The positive integers \\( n \\geq 1 \\) such that \\( n^2 + 3^n \\) is the square of an integer are \\( n = 1 \\) and \\( n = 3 \\).\n",
      "The problem provided does not contain a solution. Therefore, no final answer can be extracted.\n",
      "The possible values of $\\frac{S}{D}$ for a convex quadrilateral are all real values in the open interval $(1, 2)$.\n",
      "\\[\\frac{[ABD]}{[CDE]} = \\frac{1}{3}\\]\n",
      "The functions \\( f \\) that satisfy the given equation are:\n",
      "\\[ f(x) = 0 \\quad \\text{and} \\quad f(x) = x^2 \\]\n",
      "The number of elements of \\( S \\) is \\( 2^n \\) for any non-negative integer \\( n \\).\n",
      "The smallest integer \\( k \\) is 2022.\n",
      "It is impossible for \\( AB, AC, BI, ID, CI, IE \\) to all have integer lengths.\n",
      "The only integral solution is \\((a, b, c) = (0, 0, 0)\\).\n",
      "\\[\n",
      "\\angle C B P = \\angle B C Q = 15^\\circ\n",
      "\\]\n",
      "The composite positive integers \\( n \\) for which it is possible to arrange all divisors of \\( n \\) that are greater than 1 in a circle so that no two adjacent divisors are relatively prime are those \\( n \\) that are not of the form \\( pq \\) where \\( p \\) and \\( q \\) are distinct primes.\n",
      "The smallest integer \\( n \\) for which this is possible is \\( 23 \\).\n",
      "The sequence is \\(2, 4, 6, \\ldots, 2n-2\\).\n",
      "The functions that satisfy the given equation are:\n",
      "\\[ f(x) = 0 \\]\n",
      "and\n",
      "\\[ f(x) = x^2 \\]\n",
      "\\[\\frac{(m+n)!}{(m-n)!} = \\prod_{i=1}^{n}\\left(m^{2}+m-i^{2}+i\\right)\\]\n",
      "The sets of six consecutive positive integers that satisfy the given condition are:\n",
      "\\[ \\{2, 3, 4, 5, 6, 7\n",
      "YES\n",
      "\\begin{cases} \n",
      "1 & \\text{if } bd \\leq \\frac{ad(d+1)}{2}, \\\\\n",
      "bd - \\frac{ad(d-1)}{2} & \\text{otherwise}.\n",
      "\\end{cases}\n",
      "\\text{No solutions}\n",
      "\\text{All pairs except } (1,1), (1,2k), (2k,1)\n",
      "f(x) = a_0 (x^2 + \\alpha^2)^n \\text{ where } a_0 > 0 \\text{ and } \\alpha \\in \\mathbb{R} \\setminus \\{0\\}\n",
      "The integers \\( n > 1 \\) that have the property are \\( n \\geq 3 \\).\n",
      "\\[ P(x) = c(x^2 + 3) \\text{ for any constant } c. \\]\n",
      "The only function \\( f \\) that satisfies the given conditions is \\( f(n) = 1 \\) for all positive integers \\( n \\).\n",
      "The positive integers \\( n \\) such that there are \\( k \\geq 2 \\) positive rational numbers \\( a_1, a_2, \\ldots, a_k \\) satisfying \\( a_1 + a_2 + \\cdots + a_k = a_1 \\cdot a_2 \\cdots a_k = n \\) are:\n",
      "\\[ n = 4 \\text{ or } n \\geq 6 \\]\n",
      "The minimum number of real roots of \\( f(x) \\) is \\( 6 \\).\n",
      "The possible labels for the point $(2000, 2024)$ are precisely the multiples of 3 from 0 to 6048.\n",
      "There are no integral solutions to the given Diophantine equation.\n",
      "There is no solution provided for this problem.\n",
      "The minimal total penalty a player can amass is:\n",
      "\\[ \\min(BW, 2WR, 3RB) \\]\n",
      "\n",
      "The number of optimal strategies is:\n",
      "- If \\( BW \\neq 2WR \\neq 3RB \\), there is \\( 1 \\) optimal strategy.\n",
      "- If \\( BW = 2WR < 3RB \\), there are \\( W+1 \\) strategies.\n",
      "- If \\( 2WR = 3RB < BW \\), there are \\( R+1 \\) strategies.\n",
      "- If \\( 3RB = BW < 2WR \\), there are \\( B+1 \\) strategies.\n",
      "- If \\( BW = 2WR = 3RB \\), there are \\( R+B+W \\) strategies.\n",
      "\\[ f(x) = c(4x - a_1^2)(4x - a_2^2)\\cdots (4x - a_k^2), \\]\n",
      "where \\( a_1, a_2, \\ldots, a_k \\) are odd positive integers and \\( c \\) is a nonzero integer.\n",
      "The function \\( f(x) = kx \\) for any real value of \\( k \\).\n",
      "Yes, there does exist a subset \\( X \\) of the integers such that for any integer \\( n \\) there is exactly one solution to \\( a + 2b = n \\) with \\( a, b \\in X \\).\n",
      "\\[\n",
      "P(n+1) = \n",
      "\\begin{cases} \n",
      "\\dfrac{n}{n+2} & \\text{if } n \\text{ is even} \\\\\n",
      "1 & \\text{if } n \\text{ is odd}\n",
      "\\end{cases}\n",
      "\\]\n",
      "The intersections of \\(OA\\) and \\(OB\\) with the perpendicular to \\(PO\\) at \\(P\\).\n",
      "f(x) = 0 \\text{ or } f(x) = -x^n\n",
      "\\[ XY = R \\sqrt{3} \\]\n",
      "The points \\( P \\) on segment \\( BC \\) that satisfy the given property are such that:\n",
      "\n",
      "\\[ PB = \\frac{ab}{b+c} \\quad \\text{or} \\quad PB = \\frac{ac}{b+c}. \\]\n",
      "\\[\n",
      "\\begin{aligned}\n",
      "P_1 &= K, \\\\\n",
      "P_2 &= O, \\\\\n",
      "P_3 &= \\text{(not specified)}, \\\\\n",
      "P_4 &= I, \\\\\n",
      "P_5 &= L, \\\\\n",
      "P_6 &= G, \\\\\n",
      "P_7 &= H.\n",
      "\\end{aligned}\n",
      "\\]\n",
      "The area of triangle $OO_1O_2$ is minimized if $CX \\perp AB$.\n",
      "The real roots of the equation \\(x^4-(2\\cdot10^{10}+1)x^2-x+10^{20}+10^{10}-1=0\\) correct to four decimal places are approximately:\n",
      "\\[ x_1 \\approx 10^5 - \\frac{1 - \\sqrt{1 + 4(10^5 + 1)}}{2 \\cdot 2 \\cdot 10^5} \\]\n",
      "\\[ x_2 \\approx 10^5 + \\frac{1 + \\sqrt{1 + 4(10^5 + 1)}}{2 \\cdot 2 \\cdot 10^5} \\]\n",
      "\n",
      "Given the approximations:\n",
      "\\[ x_1 \\approx 99999.5000 \\]\n",
      "\\[ x_2 \\approx 100000.5000 \\]\n",
      "The smallest constant \\( c \\) such that \\( f(x) \\le cx \\) for every function \\( f \\) satisfying the given conditions is \\( c = 2 \\).\n",
      "The pairs of positive integers \\((x, y)\\) that satisfy the equation \\(x^y = y^{x - y}\\) are:\n",
      "\n",
      "\\[\n",
      "(x, y) = (9, 3) \\quad \\text{and} \\quad (x, y) = (8, 2)\n",
      "\\]\n",
      "The final answer is not provided as the solution is marked as \"Work In Progress\" (WIP).\n",
      "The smallest positive integer \\( N \\) for which Philena has a strategy that guarantees she can be certain of Nathan's pair after at most \\( N \\) rounds is \\( 9 \\).\n",
      "The simple modules of \\( A \\) are the 1-dimensional modules \\( S_i \\) for \\( 1 \\leq i \\leq n \\), where \\( E_{ii} \\) acts by 1 and \\( E_{ij}, E_{jj} \\) act by 0 for \\( j \\neq i \\).\n",
      "The largest integer \\( n \\) such that there exist monic quadratic polynomials \\( p_{1}(x), p_{2}(x), p_{3}(x) \\) with integer coefficients so that for all integers \\( i \\in[1, n] \\) there exists some \\( j \\in[1,3] \\) and \\( m \\in \\mathbb{Z} \\) such that \\( p_{j}(m)=i \\) is \\( \\boxed{9} \\).\n",
      "\\[\n",
      "(a, b, c, d) = (1989, 1989, 1990, 2 \\cdot 1989)\n",
      "\\]\n",
      "\\[ f(x) = 0 \\quad \\text{or} \\quad f(x) = x^2 - \\frac{1}{x} \\]\n",
      "The only possible value for the center is \\( 18 \\).\n",
      "There are no polynomials of degree 1 with rational coefficients that satisfy the given conditions.\n",
      "Cathy can win if and only if \\( n \\leq 2^{k-1} \\).\n",
      "\\[\n",
      "\\frac{AM}{MC} = \\frac{5}{3}\n",
      "\\]\n",
      "L=1 \\text{ or } L \\text{ is even}\n",
      "The smallest positive integer \\( k \\) is \\( 57 \\).\n",
      "\\prod_{i=1}^n a_i\n",
      "The polynomials \\( P \\) with integer coefficients that satisfy the given conditions are:\n",
      "\\[ P(x) = x + d \\quad \\text{for some integer } d \\geq -2022 \\]\n",
      "or\n",
      "\\[ P(x) = -x + d \\quad \\text{for some integer } d \\leq 2022. \\]\n",
      "Bob can always win.\n",
      "There are 4 possible values of \\( P \\).\n",
      "The pairs \\((P(x), Q(x))\\) of complex polynomials with leading coefficient 1 that satisfy the given conditions are:\n",
      "\\[\n",
      "(1, 1) \\quad \\text{and} \\quad (P, P+i), (P, P-i),\n",
      "\\]\n",
      "where \\(P\\) is a non-constant monic polynomial in \\(\\mathbb{C}[x]\\) and \\(i\\) is the imaginary unit.\n",
      "\\[\n",
      "\\lim_{n \\rightarrow \\infty} c_{n} = 4\\pi\n",
      "\\]\n",
      "1. When \\( t \\rightarrow +\\infty \\), \\( M(t) \\rightarrow +\\infty \\). Therefore, \\( N(t) \\) also diverges to \\( +\\infty \\).\n",
      "2. For \\( k \\neq 0 \\), \\( p_{k}(t, v) \\) decays, and \\( p(t, x, v) \\) is approaching \\( \\frac{1}{\\pi} p_{0}(t, v) \\), which is an even distribution in space.\n",
      "The very good years in the 21st century are:\n",
      "\\[ 2002, 2013, 2020, 2024, 2035, 2046, 2057, 2068, 2079. \\]\n",
      "i. Euler's formula holds; ii. Euler's formula holds\n",
      "The largest score $B$ can guarantee is \\( 4 \\).\n",
      "The only solution is \\((a, b, c, d) = (5, -3, 2, 3)\\).\n",
      "\\[\n",
      "\\text{The first player has a winning strategy for } 1346 \\text{ values of } N \\text{ between 1 and 2019 (inclusive).}\n",
      "\\]\n",
      "There are no such real numbers \\( c > 0 \\).\n",
      "\\pi / e\n",
      "b^{2}-2ac\n",
      "\\[\n",
      "\\left\\{\n",
      "\\begin{array}{l}\n",
      "f(x) = x \\\\\n",
      "f(x) = -x \\\\\n",
      "f(x) = x + c \\text{ for all rational numbers } c \\text{ iff } a = 2\n",
      "\\end{array}\n",
      "\\right.\n",
      "\\]\n",
      "The only solution is \\((p, q) = (3, 2)\\).\n",
      "The polynomials \\( f \\) with nonnegative integer coefficients such that \\( f(n) \\in K \\) whenever \\( n \\in K \\) are:\n",
      "\\[ f(n) = k \\]\n",
      "where \\( k \\in K \\), or\n",
      "\\[ f(n) = an + b \\]\n",
      "where \\( a \\) is a power of 10, \\( b \\in K \\), and \\( b < a \\).\n",
      "4^{2017}-2 \\cdot 3^{2017}+2^{2017}\n",
      "\\[ \\angle{APD} = 108^\\circ \\]\n",
      "\\(\\lim _{n \\rightarrow \\infty} \\frac{1}{n} \\mathbb{E}\\left(R_{n}\\right) = 2p - 1\\)\n",
      "The possible values of \\( x \\) are:\n",
      "\n",
      "\\[ -\\frac{\\sqrt{6}+\\sqrt{2}}{2} \\quad \\text{and} \\quad -\\frac{\\sqrt{6}-\\sqrt{2}}{2} \\]\n",
      "3^{16} / (3^{16}+1)\n",
      "\\frac{2^{2005}+1}{3 \\cdot 2^{2006}}\n",
      "\\frac{\\pi^{2}}{12}+\\frac{\\pi}{2}-2+\\ln 2\n",
      "2 \\cdot 3^{2008}\n",
      "\\frac{2013^{2014}}{2013^{2013}-1}\n",
      "2^{2013}\n",
      "2^{4012}\n",
      "\\frac{1}{2^{24}} \\text{ or } \\frac{1}{8^{8}} \\text{ or } \\frac{1}{16777216}\n",
      "\\[\n",
      "f(x) = x \\quad \\text{for all} \\; x \\in \\mathbb{R}\n",
      "\\]\n",
      "2^{1100}\n",
      "\\left(2^{2003}-2\\right) / 3\n",
      "-2^{1002}\n",
      "2017^{2015}\n",
      "30^{\\circ} \\text{ or } \\pi / 6 \\text{ radians}\n",
      "1+\\frac{1}{2^{2020}}\n",
      "4 \\cdot 3^{2011}\n",
      "\\left(\\frac{11}{18}\\right)^{2012}\n",
      "e^{23110}\n",
      "(A, C, D)\n",
      "1-\\frac{2^{15}}{3^{15}}\n",
      "x=\\frac{\\pi}{4}, \\frac{5 \\pi}{4}\n",
      "\\left(\\frac{1+2^{1005}}{2^{1007}}\\right)^{2}\n",
      "2005^{1002} / 2004!\n",
      "\\frac{2^{1009}}{2^{1009}+1009}\n",
      "2^{2014}\n",
      "101 \\cdot 2^{49}-50\n",
      "\\frac{3^{2014}}{2^{2014} \\cdot 2014!}\n",
      "\\frac{5055}{2^{15}\n",
      "\\frac{1}{2}\\left(16^{2014}-14^{2014}\\right)\n",
      "x= \\pm \\frac{\\pi}{12}, \\pm \\frac{5 \\pi}{12}\n",
      "1.813759629294 \\cdot 10^{12}\n",
      "\\binom{100}{50} 2^{98}\n",
      "\\frac{3 \\pi}{13}-\\frac{4}{13} \\log \\frac{3}{2}\n",
      "4^{2004}+2^{2004}\n",
      "1-\\ln (e-1)\n",
      "\\frac{2014\\left(1+\\binom{2013}{1007}\\right)}{2^{2014}}\n",
      "\\frac{3^{8068}-81}{80}\n",
      "\\frac{1}{2014!^{2014}}\n",
      "\\frac{2^{135}-2^{128}+1}{2^{119} \\cdot 129}\n",
      "3^{2004}-2 \\cdot 3^{1002}+2\n",
      "\\frac{3^{59}+1}{4 \\cdot 3^{59}}\n",
      "2^{32}\n",
      "\\frac{11}{2^{20}\n",
      "1 \\text{ (when } n=2\\text{); 0 \\text{ otherwise}\n",
      "\\frac{1}{2^{2012}}\n",
      "9^{36}+4\n",
      "1.01^{100}\n",
      "\\frac{-1}{2}+\\frac{(1007)\\binom{2013}{1006}}{2^{2012}}\n",
      "\\binom{\\binom{n}{k}}{m}\n",
      "n can be 1, a prime that is 1 \\bmod 3, or the square of any prime except 3\n",
      "2^{1998}\n",
      "a+b i \\text{ where } a, b \\in \\mathbb{Q} \\text{ and } \\nu_{p}(a), \\nu_{p}(b) \\geq 0 \\text{ for all } p \\equiv 1(\\bmod 4)\n",
      "1-\\left(\\frac{1}{2}\\right)^{2016}\n",
      "\\frac{A B^{2}}{A C^{2}}\n",
      "Yes\n",
      "$4^{2010}-2^{2010}$\n",
      "327680 \\cdot 2^{16}\n",
      "\\frac{4^{2024}}{\\binom{4048}{2024}}-2\n",
      "2 \\cdot\\left(3^{2017}-2^{2017}\\right)\n",
      "2n+1 \\text{ if } n \\text{ is odd, } 2n-2 \\text{ if } n \\text{ is even}\n",
      "\\frac{1}{2}\\left[\\left(\\frac{1}{3}\\right)^{100}+1\\right]\n",
      "$\\sqrt{\\frac{2^{2009}-1}{3 \\cdot 2^{2008}-1}}$\n",
      "\\left(2^{2014}-4\\right) / 3\n",
      "1, \\frac{2^{2011}+1}{3 \\cdot 2^{2011}}\n",
      "\\frac{\\binom{4030}{2016}}{2^{4030}}\n",
      "\\frac{17^{15}}{16!^{2}}\n",
      "\\[\n",
      "\\sup \\{V \\mid V \\text{ is good} \\} = \\frac{1}{4}\n",
      "\\]\n",
      "\\frac{2013^{2013}-2012^{2013}}{2013^{2012}}\n",
      "1+\\log (2016)\n",
      "2013^{4025}\n",
      "2^{1006} \\sqrt{2^{2010}+2}-2^{2011}\n",
      "3 \\mathrm{ft} / \\mathrm{s}\n",
      "2^{2007}\n",
      "\\log (2007 / 2006)\n",
      "1/3 + 1/(3 \\cdot 2^{2003})\n",
      "995 \\times 2^{998}\n",
      "\\max \\left(a_{1}, \\ldots, a_{n}, \\frac{1}{2} \\sum_{i=1}^{n}\\left|a_{i}-a_{i+1}\\right|\\right)\n",
      "(1,1) \\text{ and } \\left(2,2^{2017}\\right)\n",
      "f(n)=\\left\\{\\begin{array}{cl} 0 & \\text { if } n \\text { is even } \\\\ \\frac{1}{2 n} & \\text { if } n \\text { is odd } \\end{array}\\right.\n",
      "The maximum likelihood estimator $\\widehat{\\beta} = \\frac{1}{n} \\sum_{i=1}^{n} X_{i}$\n",
      "\\pi r \\sqrt{2}\n",
      "No positive integers n satisfy the condition.\n",
      "2^{32}\n",
      "(1009, 2^{1009}-2)\n",
      "\\frac{550}{9^{10}}\n",
      "There are two families of answers: (a) $a_{n}=c(n+2) n!$ for all $n \\geq 1$ and $a_{0}=c+1$ for some integer $c \\geq 2014$, and (b) $a_{n}=c(n+2) n!$ for all $n \\geq 1$ and $a_{0}=c-1$ for some integer $c \\geq 2016$.\n",
      "f(x)=x+c, c \\in \\mathbb{R} \\text{ constant}\n",
      "\\frac{\\left(\\left\\lfloor\\frac{2017-1337}{2}\\right\\rfloor\\right)}{2^{2017}}\n",
      "L(k)=\\begin{cases}100^{2}-(2 k-100)^{2} & \\text{if } k \\text{ is even} \\\\ \\frac{100^{2}-(2 k-100)^{2}}{2} & \\text{if } k \\text{ is odd} \\end{cases}\n",
      "All integers $n=3^{b}$, where $b$ is a nonnegative integer.\n",
      "P(x) = cx + d \\text{ with } c, d \\in \\mathbb{Z}\n",
      "k = 2^{a} \\text{ for all } a \\geq 0\n",
      "2^{2018}-1\n",
      "3^{2007}\n",
      "f(x)=x \\text{ for all positive real numbers } x\n",
      "Positions marked with $*_{i}$ in the grid\n",
      "21 for headphones, 36 for both items\n",
      "\\left(2^{2019}-1\\right)^{2018}\n",
      "f(x)=2x \\text{ for all } x>0\n",
      "Even\n",
      "Expected number: P_{1} + P_{2} + \\cdots + P_{m}; Probability: 1 - (1 - P_{1})(1 - P_{2}) \\ldots (1 - P_{m})\n",
      "S\\left(\\frac{1}{1+x^{2}}\\right)=\\pi e^{-2 \\pi|x|}, S\\left(\\frac{1}{\\left(1+x^{2}\\right)^{2}}\\right)=\\frac{\\pi}{2}(1+2 \\pi|x|) e^{-2 \\pi|x|}\n",
      "Optimal route is dynamic based on pickup\n",
      "2^{101}-1\n",
      "\\frac{7 \\cdot 2^{18}+4}{15}\\left(\\text { or } \\frac{1835012}{15}\\right)\n",
      "p_{12}^{*} = \\begin{cases} \\frac{1}{3}(c_{12} + \\sqrt{c_{12}^{2} + 6u_{1}u_{2}}), & c_{12} \\in [0, \\frac{3}{2}u_{1} - u_{2}] \\\\ \\frac{1}{4}(u_{1} + 2u_{2} + 2c_{12}), & c_{12} \\in [\\frac{3}{2}u_{1} - u_{2}, u_{2} - \\frac{1}{2}u_{1}] \\\\ \\frac{1}{3}(u_{1} + u_{2} + 2c_{12}), & c_{12} \\in [u_{2} - \\frac{1}{2}u_{1}, u_{1} + u_{2}] \\end{cases}\n",
      "c_{1}=-2 k, c_{2}=-4 \\pi^{2}\n",
      "\\frac{1}{100 \\cdot 100!^{2}}\\binom{200}{99}\n",
      "(A), (B), (C), (D)\n",
      "48 solutions by permuting vertices, adjusting sides, and exchanging middle numbers.\n",
      "f(x)=c x, c \\in \\mathbb{R}\n",
      "49!\\cdot 2^{49}\n",
      "\\frac{2^{100}}{\\binom{100}{50}}-1\n",
      "2^{2009}\n",
      "a_{1}=2^{\\ell} \\text{ for } \\ell \\geq 1 \\text{ when } m=2\n",
      "f(x)=\\frac{ax}{5} \\text{ for } x \\text{ divisible by 5 and } f(x)=bx \\text{ for } x \\text{ not divisible by 5}\n",
      "54^{48}\n",
      "p_{i}^{*} = \\frac{u_{i} + c_{i}}{2}, r_{i}^{*} = \\frac{(u_{i} - c_{i})^{2}}{4u_{i}}\n",
      "2^{2015}+\\left\\lfloor\\left(\\frac{3}{2}\\right)^{2015}\\right\\rfloor-2\n",
      "\\log _{2} 2015-1\n",
      "All powers of 2\n",
      "Two continuous arcs from the circle with diameter 4,4\n",
      "1000^{999}\n",
      "2^{2000}\n",
      "2^{2^{7}}\n",
      "2^{2015} \\cdot\\left(2^{2017}-2018\\right)\n",
      "f(x)=0 \\text{ for all } x \\text{ and } f(x)=x^{2} \\text{ for all } x\n",
      "\\frac{1}{4}+\\frac{3}{4 \\cdot 5^{2008}}\n",
      "100^{17}\n",
      "2^{35}\n",
      "$\\frac{2009\\left(2^{2010}\\right)+1}{2011!}$\n",
      "IAJCBMHODEFLNGK\n",
      "9\\left(10^{2019}-9^{2019}\\right)\n",
      "3^{671} \\cdot 4\n",
      "\\frac{2}{3}+\\frac{1}{3}\\left(\\frac{1}{4}\\right)^{2013}\n",
      "Existence of \\(\\beta_{c}\\) and limits as described in the solution.\n",
      "Joint density of \\(\\left(X_{(1)}, X_{(n)}\\right)\\), joint density of \\((R, V)\\), density of \\(R\\), and density of \\(V\\) as described in the solution.\n",
      "2^{2013}\\left(2^{2014}-1\\right) \\text{ OR } 2^{4027}-2^{2013}\n",
      "2012^{2011}\n",
      "1048577 \\text{ or } 2^{20}+1\n",
      "Yes\n",
      "70,110 \\text{ (need both, but order doesn't matter)}\n",
      "f(x) = ax + b \\text{ where } a \\in \\mathbb{Q} \\text{ and } b \\in \\mathbb{R}\n",
      "(p, q, r)=\\left(\\frac{2}{5}, \\frac{3}{10}, \\frac{3}{10}\\right)\n",
      "\\frac{507}{16384} \\text{ or } \\frac{2^{10}-10}{2^{15}} \\text{ or } \\frac{2^{9}-5}{2^{14}}\n",
      "5^{2013} \\text{ OR } 125^{671}\n",
      "(4,4,4),(-4,-4,-4) \\text{ (need both, but order doesn't matter)}\n",
      "1-\\frac{\\pi^{2}}{24} \\text{ OR } \\frac{24-\\pi^{2}}{24}\n",
      "2^{2013}-6036\n",
      "P(x, y)=0 \\text{ and } P(x, y)=\\left(x^{2}+y^{2}\\right)^{n}\n",
      "n \\text{ is even}\n",
      "7 yellow coins\n",
      "\\frac{81^{10}}{82^{10}\n",
      "2012^{2012}\n",
      "-\\pi \\log 2\n",
      "4 \\text{ (when } p > 3\\text{) or } 1 \\text{ (when } p=3\\text{)}\n",
      "f(p) \\equiv 0 \\bmod 3 \\text{ for } p=3, f(p) \\equiv 1 \\bmod 3 \\text{ for } p=3k+1, \\text{ and } f(p) \\equiv -1 \\bmod 3 \\text{ for } p=3k-1\n",
      "36 \\cdot 2^{2009}+36\n",
      "\\tan^{-1}\\left(\\frac{1009}{1005}\\right)\n",
      "Converges\n",
      "f(t)=C_{1} t+\\frac{C_{2}}{t}+C_{3}\n",
      "d(k, n) = k \\cdot n \\text{ if } k, n > 1 \\text{, otherwise } d(k, n) = k + n\n",
      "5 \\cdot 2^{2010}\n",
      "\\Delta(n, k)=\\sum_{i=0}^{k-1}\\binom{k-1}{i} \\frac{D_{(n+1)-(k+i)}}{n-(k+i)}\n",
      "(-\\infty, \\frac{1}{4}] \\cup \\{\\frac{1}{4 \\cos^{2} \\frac{k\\pi}{n}}; k \\in \\mathbb{N}, 1 \\leq k < \\frac{n}{2}\\}\n",
      "\n",
      "[\n",
      "     $$\n",
      "     f(n) = \n",
      "     \\begin{cases} \n",
      "     0 & \text{if } n \text{ is even}, \\ \n",
      "     \text{any perfect square} & \text{if } n \text{ is odd} \n",
      "     \\end{cases}\n",
      "     $$;\n",
      "     $$\n",
      "     f(n) = n^{2} \\quad \text{for every integer } n.\n",
      "     $$\n",
      "]\n",
      "\n",
      "9 \\mathrm{~km} / \\mathrm{h}\n",
      "1 \text{ p.m. Monday}\n",
      "108^{\\\\circ}\n",
      "\frac{5}{16}\n",
      "8 \\text{ p.m. on Saturday}\n",
      "\frac{3}{8}\n",
      "63 \text{ seconds}\n",
      "\frac{4}{3}\n",
      "f(x) = cx^d, c,d > 0, c=1 \\text{ if } d=1\n",
      "f(n) = \\frac{1}{9}(10^c (9n+1)^d - 1) for c \\geq 1-d, d \\geq 0\n",
      "n=2^k-1 \\text{ for some integer } k\\geq 1\n",
      "All real numbers $a > 2$.\n",
      "The coordinates of $R(x,y)$ are $(x+n,y)$.\n",
      "-\\frac{3^{1010}-1}{2} \\text{ and } -\\frac{3^{1010}-1}{2}\\pm\\frac{\\sqrt{9^{1010}-1}}{4}\\,i\n",
      "The values of $n$ are $2^{2^1}$, $2^{2^2}$, $2^{2^4}$, $2^{2^8}$.\n",
      "c = -1, L = \\frac{2}{\\pi}\n",
      "\\frac{\\pi \\log(2)}{8}\n",
      "$(\\log 2)^2$\n",
      "j \\text{ not divisible by either } 42 \\text{ or } 46\n",
      "3m^2 \\text{ for some positive integer } m\n",
      "n$ is odd\n",
      "f(x) = cx+d\n",
      "Such a set exists for every $n \\geq 1.\n",
      "$(a,b,c) = (9,8,7)$\n",
      "Multiples of 3 starting with 9\n",
      "p(x) = ax+b, q(x) = cx+d \\text{ with } bc-ad=1\n",
      "At most two rational points.\n",
      "f(x) = x^c \\text{ for some } c>0\n",
      "The answer is no.\n",
      "The length of the altitude of the tetrahedron from vertex \\(A\\) can be constructed as \\(\\sqrt{|A_BY'|^2 - |X'Y'|^2}\\).\n",
      "The probability that $C$ is a polygon with four vertices is larger than the probability that $C$ is a polygon with three vertices.\n",
      "\\[\n",
      "\\boxed{\\text{prime } n}\n",
      "\\]\n",
      "\\[\n",
      "\\begin{array}{ll}\n",
      "\\text{(a)} & \\text{No} \\\\\n",
      "\\text{(b)} & \\text{Yes}\n",
      "\\end{array}\n",
      "\\]\n",
      "The problem provided does not contain a solution. Therefore, no final answer can be extracted.\n",
      "Player B has a winning strategy for \\( n \\geq 4 \\).\n",
      "The game is winnable if and only if \\( n > k \\ge 2 \\).\n",
      "s_{1}=3, s_{2}=2, \\text{arrows on all even-numbered squares}\n",
      "\\[ AP \\cdot PB = 4rR \\cos^2 \\alpha \\]\n",
      "|V|^{2016}\n",
      "\\frac{1}{2^{4016}}\n",
      "\\frac{\\sqrt{4009}}{2^{2004}}\n",
      "\\frac{2^{17}}{125}\n",
      "1-2^{2012}\n",
      "90 \\text{ mph}\n",
      "5^{56}, 31^{28}, 17^{35}, 10^{51}\n",
      "A responds after \\frac{n-1}{2} 'no' responses if n is odd, after \\frac{n}{2} 'no' responses if n is even\n",
      "Yes\n",
      "less than 50\n",
      "$x^{9900}\\left(\\frac{1+x^{100}}{2}\\right)^{2}$ OR $\\frac{1}{4} x^{9900}+\\frac{1}{2} x^{10000}+\\frac{1}{4} x^{10100}$\n",
      "\\left(\\frac{m^{3}+6 m^{2}+11 m+6}{6}\\right)^{3} \\text{ OR } \\binom{m+3}{3}^{3}\n",
      "\\log _{2008} 2009\n",
      "x = \\alpha if n is odd, x = \\alpha or \\alpha if n is even\n",
      "Throughout this problem, we will assume that the given circles are externally tangent, since the problem does not have a solution otherwise. Let $\\Gamma_{1}$ and $\\Gamma_{2}$ be the given circles and $T$ be their tangency point. Suppose $\\omega$ is a circle that is tangent to $\\Gamma_{1}$ and $\\Gamma_{2}$ and passes through $P$. Now invert about point $P$, with radius $PT$. Let any line through $P$ that cuts $\\Gamma_{1}$ do so at points $X$ and $Y$. The power of $P$ with respect to $\\Gamma_{1}$ is $PT^{2}=PX \\cdot PY$, so $X$ and $Y$ are swapped by this inversion. Therefore $\\Gamma_{1}$ is mapped to itself in this inversion. The same applies to $\\Gamma_{2}$. Since circle $\\omega$ passes through $P$, it is mapped to a line tangent to the images of $\\Gamma_{1}$ (itself) and $\\Gamma_{2}$ (also itself), that is, a common tangent line. This common tangent cannot be $PT$, as $PT$ is also mapped to itself. Since $\\Gamma_{1}$ and $\\Gamma_{2}$ have exactly other two common tangent lines, there are two solutions: the inverses of the tangent lines. We proceed with the construction with the aid of some macro constructions that will be detailed later. Step 1. Draw the common tangents to $\\Gamma_{1}$ and $\\Gamma_{2}$. Step 2. For each common tangent $t$, draw the projection $P_{t}$ of $P$ onto $t$. Step 3. Find the inverse $P_{1}$ of $P_{t}$ with respect to the circle with center $P$ and radius $PT$. Step 4. $\\omega_{t}$ is the circle with diameter $PP_{1}$. Let's work out the details for steps 1 and 3. Steps 2 and 4 are immediate. Step 1. In this particular case in which $\\Gamma_{1}$ and $\\Gamma_{2}$ are externally tangent, there is a small shortcut: - Draw the circle with diameter on the two centers $O_{1}$ of $\\Gamma_{1}$ and $O_{2}$ of $\\Gamma_{2}$, and find its center $O$. - Let this circle meet common tangent line $OP$ at points $Q, R$. The required lines are the perpendicular to $OQ$ at $Q$ and the perpendicular to $OR$ at $R$. Let's show why this construction works. Let $R_{i}$ be the radius of circle $\\Gamma_{i}$ and suppose without loss of generality that $R_{1} \\leq R_{2}$. Note that $OQ=\\frac{1}{2}O_{1}O_{2}=\\frac{R_{1}+R_{2}}{2}, OT=OO_{1}-R_{1}=\\frac{R_{2}-R_{1}}{2}$, so $$\\sin \\angle TQO=\\frac{OT}{OQ}=\\frac{R_{2}-R_{1}}{R_{1}+R_{2}}$$ which is also the sine of the angle between $O_{1}O_{2}$ and the common tangent lines. Let $t$ be the perpendicular to $OQ$ through $Q$. Then $\\angle(t, O_{1}O_{2})=\\angle(OQ, QT)=\\angle TQO$, and $t$ is parallel to a common tangent line. Since $$d(O, t)=OQ=\\frac{R_{1}+R_{2}}{2}=\\frac{d(O_{1}, t)+d(O_{2}, t)}{2}$$ and $O$ is the midpoint of $O_{1}O_{2}, O$ is also at the same distance from $t$ and the common tangent line, so these two lines coincide. Step 3. Finding the inverse of a point $X$ given the inversion circle $\\Omega$ with center $O$ is a well known procedure, but we describe it here for the sake of completeness. - If $X$ lies in $\\Omega$, then its inverse is $X$. - If $X$ lies in the interior of $\\Omega$, draw ray $OX$, then the perpendicular line $\\ell$ to $OX$ at $X$. Let $\\ell$ meet $\\Omega$ at a point $Y$. The inverse of $X$ is the intersection $X^{\\prime}$ of $OX$ and the line perpendicular to $OY$ at $Y$. This is because $OYX^{\\prime}$ is a right triangle with altitude $YX$, and therefore $OX \\cdot OX^{\\prime}=OY^{2}$. - If $X$ is in the exterior of $\\Omega$, draw ray $OX$ and one of the tangent lines $\\ell$ from $X$ to $\\Omega$ (just connect $X$ to one of the intersections of $\\Omega$ and the circle with diameter $OX$). Let $\\ell$ touch $\\Omega$ at a point $Y$. The inverse of $X$ is the projection $X^{\\prime}$ of $Y$ onto $OX$. This is because $OYX^{\\prime}$ is a right triangle with altitude $YX^{\\prime}$, and therefore $OX \\cdot OX^{\\prime}=OY^{2}$.\n",
      "HMPCVSE\n",
      "(\\ell, r) \\text{ satisfies the required conditions if and only if } \\ell \\equiv r \\equiv 1 \\text{ or } \\ell \\equiv r \\equiv 3 \\pmod{4}\n",
      "c_{1} \\sqrt{n} \\leq K P_{n} \\leq c_{2} \\sqrt{n}\n",
      "F, G, A, D, E, B, C \\text{ OR } F<G<A<D<E<B<C \\text{ OR } C>B>E>D>A>G>F\n",
      "If there were 60 spectators, then it is possible that all spectators could see the entire show, but if there were 800 spectators, then some of them could not see the entire show.\n",
      "1-\\frac{\\sin ^{2}\\left(2^{2011} x\\right)}{4^{2011} \\sin ^{2}(x)}\n",
      "2 \text{ hours and } 13 \text{ minutes}\n",
      "7 \times 2023\n",
      "Bev\n",
      "Tuesday\n",
      "a^{2}<a<\frac{1}{a}\n",
      "Doing some yoga\n",
      "W, Y, Z\n",
      "Thursday\n",
      "15 \\mathrm{~km} / \\mathrm{h}\n",
      "Bev\n",
      "Jie\n",
      "Friday\n",
      "Sunday\n",
      "89 \\mathrm{~km} / \\mathrm{h}\n",
      "62 \\mathrm{~km} / \\mathrm{h}\n",
      "48 \\mathrm{~km} / \\mathrm{h}\n",
      "Graph Q\n",
      "\frac{1}{2}\n",
      "\frac{7}{12}\n",
      "Yes, $A_1 \\le A_2$.\n",
      "Primes 2 and 5 appear seven or more times.\n",
      "0 wins, 1 loss, 2 ties\n",
      "For all $n$, Bob has a winning strategy.\n",
      "The sum diverges.\n",
      "No, there do not exist such polynomials.\n",
      "Converges for \\(b=2\\); diverges for \\(b \\geq 3\\)\n",
      "4:45 \text{ p.m.}\n",
      "Yes, the maximum length is about 4.0027.\n",
      "Player 0 wins\n",
      "Integers with no $0$'s in their base 10 expansion.\n",
      "\\log \\frac{4}{3}\n",
      "m is either a square or twice a square.\n",
      "3^{2021} - 2^{2021}\n",
      "f(x) = \\alpha - x \\text{ for some constant } \\alpha.\n",
      "\\text{All positive integers } n \\text{ with prime factors } 1 \\pmod{4}.\n",
      "f(x) = 0 \\text{ for all } x \\in \\mathbb{R}\n",
      "f(x) = x \\text{ and } P(x) = x\n",
      "The number of distinct such colorings of \\( P \\) is \\(\\boxed{6}\\).\n",
      "\\text{even integers } n \\geq 2\n",
      "\\text{Player A can win for sure}\n",
      "\\text{No such integer } a.\n",
      "\\text{m is not squarefree}\n",
      "There are no pairs \\((m, n)\\) of integers that satisfy the equation \\(m^5 - n^5 = 16mn\\).\n",
      "\\{ f \\in \\mathbb{R}[x] \\colon 2 \\mid \\deg f + 1 \\}\n",
      "2^{101} - 1\n",
      "f(x) = q^{d(x)-1}\n",
      "(m, n) \\text{ such that } m \\leq n + 1.\n",
      "f(n) = 1 \\text{ for all } n \\in \\mathbb{N} \\text{ or } f(n) = n^2 \\text{ for all } n \\in \\mathbb{N}.\n",
      "(a, b, c) = (a, 0, a) \\text{ or } \\left( \\frac{b^2 - b}{2}, b, \\frac{b^2 + b}{2} \\right)\n",
      "(a, b, p, n) = (2^k, 2^k, 2, 2013k + 1)\n",
      "\\frac{1}{2^{1007}}\n",
      "$(p,q,r,k)= (2, 3, 11, 5); (2, 11, 3, 5); (3, 11, 2, 5); (3, 2, 11, 5); (11, 2, 3, 5); (11, 3, 2, 5) $\n",
      "The final answer is not provided in the given solution.\n",
      "\\text{Yes}\n",
      "\\arcsin \\frac{1}{\\sqrt{3}}\n",
      "f(x) = 0, \\quad f(x) = x, \\quad \\text{and} \\quad f(x) = -x.\n",
      "f(x) = 1 \\text{ for all } x \\in \\mathbb{R}\n",
      "f(x) = Ax + B \\text{ for some } A, B \\in \\mathbb{Q}.\n",
      "\\text{Bob has a winning strategy.}\n",
      "(a, 0) \\text{ for any real number } a.\n",
      "\\frac{y \\pm \\sqrt{y^2 - 4}}{2} \\text{ where } y \\text{ is an integer with } |y| \\geq 2\n",
      "\\text{Anna has a winning strategy if and only if } k \\text{ is not the smallest of the } k \\text{ topmost cards.}\n",
      "\\text{All points } P \\text{ inside the triangle such that } \\angle APB = 120^\\circ, \\angle BPC = 120^\\circ, \\text{ or } \\angle CPA = 120^\\circ.\n",
      "n = 2^a \\text{ for integer } a > 1\n",
      "f(x) = x + 1 \\text{ and } g(y) = y + 1\n",
      "f(x) = x \\text{ for all } x \\in \\mathbb{R}\n",
      "\\text{Charlie}\n",
      "f(n) = n^c \\text{ for some } c \\in \\mathbb{R}\n",
      "f(x) = \\frac{x^3 - x}{3} + tx \\text{ for any } t \\in \\mathbb{Z}.\n",
      "f(x) = x^2 + cx \\text{ for any integer } c.\n",
      "\\text{The disc with diameter } AB \\text{ excluding its center.}\n",
      "\\text{Yes}\n",
      "\\frac{ab}{c}\n",
      "2^{2022} - 1\n",
      "f(x) = ax \\text{ for some integer } a.\n",
      "(x, y, z, t) = (3, 1, 2, 2)\n",
      "(p, x, y) = (p, p, 0) \\text{ or } (p, 0, p)\n",
      "(-1)^n 2^{n-1} d^n (2a + nd)\n",
      "2 \\text{ if } n = 2k, \\text{ and } 1 \\text{ otherwise}\n",
      "2n - 3 \\text{ if } n \\text{ is odd, and } 2n - 4 \\text{ if } n \\text{ is even}\n",
      "g(n) = n + c \\text{ for some } c \\in \\mathbb{Z}_{\\ge 0}.\n",
      "f(a) = ka \\text{ for any positive integer } a \\text{ and some positive integer } k.\n",
      "P(x) = cx\n",
      "$\\{(n,n) \\colon n \\in \\mathbb{Z}\\} \\cup \\{(0,7), (12,3), (-18,-2)\\}.$\n",
      "9k \\text{ for any positive integer } k.\n",
      "\\lfloor \\log_2(n) \\rfloor + 1\n",
      "f(x) = 1 \\text{ for all } x \\in \\mathbb{Q}_{>0}.\n",
      "f(x) = 0 \\text{ for all } x \\in \\mathbb{Z} \\quad \\text{and} \\quad f(x) = x + 1 \\text{ for all } x \\in \\mathbb{Z}.\n",
      "\\angle BEA_1 = 90^\\circ \\text{ and } \\angle AEB_1 = 90^\\circ\n",
      "f(x) = c \\text{ for } c \\in \\mathbb{Z}, \\quad f(x) = \\lfloor x \\rfloor, \\quad f(x) = \\lceil x \\rceil\n",
      "\\text{All prime numbers}\n",
      "f(n) = \\prod_{i=1}^k p_i^{p_i^{\\alpha_i} - 1}\n",
      "(x,y,z)=\\left(\\frac{b+c}{2},\\frac{a+c}{2},\\frac{a+b}{2}\\right)\n",
      "f(x) = C, \\quad f(x) = \\pm x + C, \\quad \\text{or} \\quad f(x) = \\pm x^3 + C\n",
      "{\\text{all pairs } (m,n)\\text{ such that } m \\nmid n,n \\nmid m.}\n",
      "f(x) = e x^a + c \\text{ where } a \\mid n \\text{ and } |e| = 1.\n",
      "f(x) = T \\cdot \\prod_{i=1}^{m} (4x - a_i)\n",
      "3k \\text{ for integers } k \\geq 1\n",
      "m \\mid n \\text{ with } n > m \\text{ and } n \\geq 3.\n",
      "The minimum number of operations is going to be $n$. \n",
      "The maximum number is $3n-2$.\n",
      "\\lceil \\log_a d \\rceil\n",
      "\\text{All pairs } (m, n) \\text{ of nonzero integers such that } \\gcd(m, n) = 1.\n",
      "C \\ge 2 \\ln 2\n",
      "minimum $4N$, maximum $(N+1)^{2}+N^{2}$.\n",
      "\\text{a circle}\n",
      "$\\boxed{(x,y,z) = (2,1,1)}$\n",
      "f(x) = 0 \\text{ and } f(x) = x^2 + a \\text{ for some real constant } a.\n",
      "floor[n/2](n-(1+floor[n/2]))\n",
      "3^{100}\n",
      "{f(x)=2 \\; \\forall x \\in \\mathbb R}\n",
      "$P(x)=x^{k} \\text{ if }n\\text{ is even, and if }n \\text{ is odd then }P(x)=-x^{k}$\n",
      "(a, 2^k - a) \\text{ for odd } a \\text{ and positive } k\n",
      "{f(x)=0\\text{ if } x\\ge 0 \\text{ and } f(x) = 2x \\text{ if }x<0}\n",
      "a=b=c=d\n",
      "n = p^m \\text{ for some prime } p \\text{ and integer } m \\geq 2.\n",
      "$abc=0$\n",
      "{m = 2^{2006}\\left(2s+1\\right)+1}\n",
      "(a_1, a_2, \\ldots, a_{n-1}) = \\left(1(2^n - 1) - (2^1 - 1)m, 2(2^n - 1) - (2^2 - 1)m, \\ldots, (n-1)(2^n - 1) - (2^{n-1} - 1)m \\right)\n",
      "$n \\equiv 1, 5 \\ ( \\text{mod} \\ 6) \\text { except } 5 \\text {and } 17$\n",
      "x_i = 2i \\text{ for } i = 1, 2, \\ldots, n-1.\n",
      "(2, 2), (3, 3), (1, p) \\text{ for any prime } p\n",
      "$f(x) = x^n, \\ n \\in \\mathbb{Z}^+ , \\ \\text{and} \\ f(x) = q^m, \\ q \\in \\mathbb{P}, \\ m \\in \\mathbb{Z}^+$\n",
      "P(x) = a_1 x + a_0 \\text{ where } a_1, a_0 \\in \\mathbb{Q} \\text{ and } a_1 \\neq 0.\n",
      "$\\boxed{f(x)=cx},\\text{其中} c \\in \\mathbb{R}$\n",
      "$f(x)=0, f(x)=-x+k$\n",
      "\\text{The sequence is such that the polynomial } P(x) = a_{0} + a_{1} x + \\ldots + a_{n} x^{n} \\text{ has only real roots.}\n",
      "m \\text{ is a prime number}\n",
      "\\begin{itemize}\n",
      "    \\item In turn 0, Susana writes 1.\n",
      "    \\item In turn 1, suppose Brenda writes $nx \\pm 1$. As $n \\neq \\pm 1$ (by rules of the game), this polynomial does not have an integer root.\n",
      "    \\item In turn 2, if Brenda wrote $nx - 1$, Susana writes $(n + 1)x^2 + nx - 1$. As $-1$ is a root of this polynomial, Susana wins.\n",
      "    Similarly, if Brenda wrote $nx + 1$, Susana writes $(n + 1)x^2 - nx - 1$. As 1 is a root, she wins too.\n",
      "\\end{itemize}\n",
      "m = 2 \\text{ and } a_1 \\text{ is a power of two}\n",
      "\\text{Beto wins}\n",
      "f(x) = x; \\quad f(x) \\equiv 0; \\quad f(x) = \n",
      "\\begin{cases} \n",
      "0, & x \\neq -a^2 \\\\\n",
      "a, & x = -a^2 \n",
      "\\end{cases}\n",
      "\\text{ for arbitrary } a \\in (-\\infty, -1] \\cup (0, +\\infty).\n",
      "n-3 \\text{ if } n \\equiv 1 \\pmod{2}, \\text{ and } n-2 \\text{ otherwise}.\n",
      "$P(x)=\\pm x^d+c \\text{, where } c \\text {is an integer and }d\\text{ is a positive integer.}$\n",
      "\\text{The perpendicular bisector of segment } BC.\n",
      "f(x) = x \\text{ for all } x \\in \\mathbb{R}f(x) = -x \\text{ for all } x \\in \\mathbb{R}\n",
      "f(x) = x \\quad \\text{and} \\quad g(x) = x.\n",
      "m \\text{ is odd}\n",
      "(2^k, 2^k), (2 \\cdot 3^k, 3^k), (3^k, 2 \\cdot 3^k) \\text{ for non-negative integers } k.\n",
      "\\text{For } n = 3, \\text{ it is not possible. For } n = 2, \\text{ it is possible.}\n",
      "(a, b, c) \\text{ such that the ratios of any two are not perfect cubes}\n",
      "\\text{ when } m \\ge 5, n \\ge 4\\text{ Alice has a winning strategy, otherwise Bob has }\n",
      "\\text{Further combinatorial analysis required}\n",
      "{\\{x_1,x_2,……….,x_{p+r-1} \\}=\\{ \\underbrace{1,1,………,1}_{r},\\underbrace{0,0,0,0,0……}_{p-1}\\}}\n",
      "n \\text{ is even}\n",
      "f(x) = c \\text{ for any } c > 0.\n",
      "$\\text { All rectangles. }$\n",
      "f(x) = ax \\text{ for some } a > 2.\n",
      "f(x) = c \\log(x) \\text{ or } f(x) = cx^k.\n",
      "$\\text{All polynomials of the form } f(x)=x^m \\text{ for some } m \\in \\mathbb{Z}^+ \\text{ and } f(x)=c \\text{ for some } c \\in \\mathbb{Z}^+ \\text{ with } \\omega(c) \\leq 2023^{2023}+1$\n",
      "\\text{even integers } n \\geq 2\n",
      "$\\textbf{All even numbers are blue}$\n",
      "m= \\left\\{ \\begin{array}{lr} 2k+3 & a=b=2k,k\\in \\mathbb{N} \\\\ \\max(a,b)+2 & \\text{otherwise} \\end{array} \\right.\\\n",
      "\\text{a) Yes, b) No}\n",
      "$\\text{ Odd numbers and powers of 2 }$\n",
      "\\boxed{(k!-1,1,1)}\\text{ (and its permutations), where }k\\in\\mathbb{N}_{>1}\n",
      "\n",
      "\\begin{cases} \n",
      "\\text{No solution} & \\text{if } p < 23, \\\\\n",
      "(3p, 3p, 3p) & \\text{if } p = 23, \\\\\n",
      "(3p, 3p, 3p), (4p, 4p, 2p), (4p, 2p, 4p), (2p, 4p, 4p) & \\text{if } p = 29, \\\\\n",
      "(3p, 3p, 3p), (4p, 4p, 2p), (4p, 2p, 4p), (2p, 4p, 4p), (6p, 3p, 2p), (6p, 2p, 3p), (2p, 3p, 6p), (2p, 6p, 3p), (3p, 2p, 6p), (3p, 6p, 2p) & \\text{if } p \\geq 31.\n",
      "\\end{cases}\n",
      "\n",
      "n - 32q - r - 1\n",
      "(n, k) \\text{ such that } k = 0 \\text{ or } k = n, \\text{ or both } n \\text{ and } k \\text{ are even, \\text{ or } (n, k) = (2, 1).}\n",
      "[1, \\infty)\n",
      "P(x,y)=(x-2y)(x+y)^{n-1}\n",
      "$\\text{第二个玩家有获胜策略，当且仅当} \\displaystyle n \\leq \\varphi k \\text{和}  \\displaystyle k \\leq \\varphi n \\text{而且}  \\displaystyle \\varphi=\\frac{\\sqrt{5}+1}{2}$\n",
      "n=kp^{2}\n",
      "$\\boxed{2^{19} \\cdot 3^{10} \\cdot 5^{4} \\cdot 3^{3} \\cdot 11\\cdot 13\\cdot 17\\cdot 19}$\n",
      "\\frac{EG}{EF}=\\frac{t}{1-t}\n",
      "{\\frac{2\\left(\\sin\\frac{\\pi}{3\\cdot 2^{99}}\\right)\\left(1-\\cos\\frac{\\pi}{2^{98}}\\right)}{\\sin \\frac{\\pi}{2^{99}}}}\n",
      "$\\text{ all odd n } , n \\geq 3$\n",
      "f(x) = \\frac{1}{x + a}\n",
      "\\text{The assistant encodes the permutation of the } k \\text{ balls using their lexicographic index and positions the block accordingly. The illusionist decodes the position to determine the permutation.}\n",
      "f(x) = 1 \\text{ for all } x \\in \\mathbb{N}.\n",
      "{\\text{all numbers that are not powers of 2 greater than 1.}}\n",
      "\\text{(a) Yes, (b) No}\n",
      "f(x) = a \\nu_p(x)\n",
      "{f(x) \\equiv 0, g(x) \\equiv 0} \\text{ or } {f(x) \\equiv x^2+c, g(x) \\equiv x}\n",
      "P(x)=\\alpha x^4+\\beta x^2,\\text{for all real number } \\alpha \\text{ and } \\beta\n",
      "\\[ n = 2^a(2^b+1) \\text{ where } a, b \\text{ are nonnegative integers not both zero.} \\]\n",
      "$\\boxed{f(n)=n+c},\\boxed{f(n)\\equiv 1},\\boxed{f(even)=1, f(odd)=2},\\boxed{f(odd)=1,f(even)=2}$\n",
      "f(x)=0\\forall x\\in\\mathbb{R},f(x)=c\\forall x\\in\\mathbb{R}, 1\\leq c<2\n",
      "P(x) = a(rx + s)^d \\ \\text{where} \\ a, r, s \\ \\text{are integers with} \\ a \\neq 0, r \\geq 1 \\ \\text{and} \\ (r, s) = 1.\n",
      "\\text{No solution}\n",
      "\\[\n",
      "(a, b) = (2l, 1) \\quad \\text{or} \\quad (l, 2l) \\quad \\text{or} \\quad (8l^4 - l, 2l)\n",
      "\\]\n",
      "for some positive integer \\( l \\).\n",
      "\\text{All non-decreasing arithmetic sequences of positive integers.}\n",
      "(a,b,c)=(2,2,2), (2,2,3), (2,6,11), (3,5,7)\n",
      "(x,y) = (11,1), (49,1), (7t^2,7t), t \\text{ is an interge}\n",
      "\\[\n",
      "\\boxed{\n",
      "\\begin{cases}\n",
      "1 & \\text{if }n\\text{ is a power of }2 \\\\\n",
      "2 & \\text{otherwise}\n",
      "\\end{cases}\n",
      "}.\n",
      "\\]\n",
      "P(x)=c\\text{ where } c\\in\\{1,...,9\\}\\text{ as well as } P(x) = x\n",
      "{f(x)=ax+\\tfrac bx}\n",
      "\\text{rational}\n",
      "\\text{all even positive integers}\n",
      "{f(x)=c(x-|x|)\\text{ }\\forall x}\n",
      "\\text{Player } B \\text{ has a winning strategy.}\n",
      "\\text{Three weighings are sufficient to find the two counterfeit coins.}\n",
      "\\frac{n+1}{n}, \\text {for any nonzero integer } n\n",
      "f(x)=k\\left(x+\\frac{1}{6}\\right)^2\n",
      "(p, x, y) \\in \\{(3, 2, 5), (3, 5, 2)\\} \\cup \\{(2, n, 2^k - n) \\mid 0 < n < 2^k\\}\n",
      "\\frac{S_{\\triangle ABC}}{4}\n",
      "\\text{All odd integers } n \\geq 3.\n",
      "（a，b，n，p）=（3，0，3，3）\n",
      "/text{yes}\n",
      "$ (x,y,z)=(1,2,5),(1,3,5),(2,2,4),(2,6,4)$\n",
      "{\\text{no solutions}}\n",
      "\\text{All pairs } (u, v) \\text{ of distinct integers for which } u \\neq 0, v \\neq 0, \\max(u, v) > 0, \\text{ and } (u, v) \\notin \\{(-1, 1), (1, -1)\\}.\n",
      "$\\boxed{f(x)=f(1)\\cdot x \\mid x\\in S; f(x) = 0 \\mid x\\not\\in S}$\n",
      "\\text{  all prime numbers}\n",
      "n=3^{1991}\n",
      "{f(x,y,z)=\\frac{3xyz}{x+y+z}}\n",
      "$\\angle CAB=60^{\\circ}$\n",
      "$f(x) = t(x-t)/(t+1), g(x) = t(x-t) \\text{ for any real t not equal to -1.}$\n",
      "\\text{The points } P, Q, \\text{ and } R \\text{ are the intersections of the line segment } M_2M_3 \\text{ with the sides } AB, BC, \\text{ and } AC \\text{ respectively.}\n",
      "$\\text { We conclude that } I\\star (M\\star O) = (O\\star I)\\star M \\text { if } \\triangle IMO \\text{ is positively oriented and is isosceles with } OI = OM \\text { and } \\angle IOM=\\frac{2\\pi}{3}.$\n",
      "\\text{Ingrid for } p = 3 \\text{ and } p = 5, \\text{ Draw for } p = 7, \\text{ Erik for } p > 7.\n",
      "\\text{For }n\\in \\{1,2,4,6\\}\\text{ the game ends in a draw, else }B\\text{ wins}\n",
      "$\\text{All composite numbers together with 2.}$\n",
      "$ a\\in\\{\\sqrt{\\frac{\\alpha}{\\beta}}\\}$\n",
      "\\text{Either} \\ (x,y) = (1,1) \\ \\text{or} \\ \\{x,y\\} = \\{m^3 + m^2 - 2m - 1, m^3 + 2m^2 - m - 1\\} \\ \\text{for some positive integer} \\ m \\geq 2.\n",
      "$f(n) = n + 1 \\text{ for all n; or, for some } a \\ge 1 ,f(n) = \\left\\{\\begin{matrix}\n",
      "  n + 1,&n > -a,\\\\-n + 1, \n",
      "  & n \\le -a;\n",
      "\\end{matrix}\\right. \\text{ or } f(n) = \\left\\{\\begin{matrix}\n",
      "  n + 1,&n > 0, \\\\\n",
      "  0,&n = 0, \\\\\n",
      "  -n + 1,&n < 0.\n",
      "\\end{matrix}\\right. $\n",
      "\\[ (x_1,x_2,\\ldots,x_{2011})=(1,k,\\ldots,k) \\text{ with } k=2+3+\\cdots+2011=2023065.\\]\n",
      "$f(x) = x,f(x) = 0 \\text{ and all functions of the form } f(x) =\\left\\{\\begin{matrix}\n",
      "  1,&x \\notin X, \\\\\n",
      "  -1,&x \\in X,\n",
      "\\end{matrix}\\right. \\text{ where } X \\subset (-\\infty , \\frac{-2}{3} ) $\n",
      "x=\\frac{1}{2}\\pm \\frac {\\sqrt 3} {2} i, n = 2+6k\\ (k\\ge 0)\n",
      "f(n) = n \\text{ for all } n \\in \\mathbb{N}\n",
      "\\[ y = \\frac{1}{m - 1} \\cdot \\left(b - \\frac{S}{m + 2}\\right),\\ x = \\frac{1}{m - 1} \\cdot \\left(c - \\frac{S}{m + 2}\\right),z = \\frac{1}{m - 1} \\cdot \\left(a - \\frac{S}{m + 2}\\right).  \\]\n",
      "{(a,1,n),(1,m,n)} \\text{ and }{(2,3,n)\\text{ where }n>1}\n",
      "$1\\text{ for } n=1\\text{ and } n^4-n^3\\text{ for }n \\ge 2$\n",
      "$\\boxed{f(x)=(x^2+1)^n},n\\in\\mathbb N_0$\n",
      "\\text{iff }k\\text{ is a prime}\n",
      "$n = 2^t \\text{ and } k \\le 2^t ? 1.$\n",
      "\\[\n",
      "\\text{For all } n \\geq 2.\n",
      "\\]\n",
      "{(a,b)\\in\\{(0,1),(1,0)\\}\\cup\\left(\\bigcup_{k\\in\\mathbb N}\\{(f_k,f_{k+1})\\}\\right)}\n",
      "\\boxed{(\\pm 1,x,\\frac{1}{x})}\\text{ and permutations.}\n",
      "{x \\in \\mathbb{Z} \\cup \\bigcup_{n = 1} ^{\\infty} (n, \\sqrt{(n-1)^2 + 1} + 1)}\n",
      "P\\text{ is the circumcenter of }\\triangle{ABC}\n",
      "(x, y, z)=(15, 9, -16)\\text{ and its permutations}\n",
      "$\\boxed{(1,2,1,0), (2,0,2,0), (2,1,2,0,0), (x-3,2,1,0,0,\\ldots, 0,1,0,0,0)} \\text{ for any } x \\ge 6$\n",
      "\\text{All prime numbers or } n = 1\n",
      "\\text{ prime } n \\text{ and }n=2^k\n",
      "f(t) = 0 \\text{ for all } t.\n",
      "\\text{ OR }\n",
      "f(t) = 0 \\text{ for } t \\text{ even and } f(t) = f(1) \\text{ for } t \\text{ odd}\n",
      "\\text{ OR }\n",
      "f(t) = 4f(1) \\text{ for } t \\text{ even and } f(t) = f(1) \\text{ for } t \\text{ odd}\n",
      "\\text{ OR }\n",
      "f(t) = t^2 f(1) \\text{ for any } f(1).\n",
      "\\tan(\\sin x) > \\sin(\\tan x) \\text{ for } x \\in \\left(0, \\frac{\\pi}{2}\\right)\n",
      "\\[\n",
      "a_{u+vn} = \n",
      "\\begin{cases} \n",
      "0, & u+v \\le n, \\\\ \n",
      "1, & u+v \\ge n+1 \n",
      "\\end{cases} \n",
      "\\quad \\text{for all } 1 \\le u \\le n \\text{ and } 0 \\le v \\le n.\n",
      "\\]\n",
      "\\[\n",
      "\\text{The terms can be arranged into blocks of length } n \\text{ as}\n",
      "\\]\n",
      "\\[\n",
      "\\underbrace{(0 \\cdots 0)}_{n} \\underbrace{(0 \\cdots 0 \\ 1)}_{n-1} \\underbrace{(0 \\cdots 0 \\ 1 \\ 1)}_{n-2} \\cdots \\underbrace{(0 \\cdots 0 \\ 1 \\cdots 1)}_{n-v} \\underbrace{(0 \\ 1 \\cdots 1)}_{v} \\cdots \\underbrace{(0 \\ 1 \\cdots 1)}_{n-1} \\underbrace{(1 \\cdots 1)}_{n}.\n",
      "\\]\n",
      "$\\text{ All even integers satisfy the condition of the problem and no other real number α does so. }$\n",
      "\\text{Alice}\n",
      "k = \\lfloor \\log_2 n \\rfloor + 1\n",
      "f(x)=x^k+\\frac{1}{x^k}\n",
      "\\text{The probability that } C \\text{ is a quadrilateral is larger than the probability that } C \\text{ is a triangle.}\n",
      "f(x) = ax^m\\text{ for some nonnegative integers } a \\text{ and }  m\n",
      "f(n)=n+1,\\ f(n)=\\begin{cases}n+1,\\ n=2k\\\\\n",
      "n+5,\\ n=4k+1\\\\\n",
      "n-3,\\ n=4k+3\n",
      "\\end{cases}\n",
      "(a, b, c) = (10, 10, 1), (10, 1, 10), (1, 10, 10)\n",
      "\\text{Bella's tour will surely cost at least as much as Anna's tour.}\n",
      "f(x) = 2x - 1, \\quad f(x) = x^2 - 1, \\quad \\text{and} \\quad f(x) = -x - 1.\n",
      "\\text{Yes}\n",
      "\\text{the medial triangle}\n",
      "{(x,y,z) = (0,0,0), (1,2,3), (4,8,6)}\n",
      "[\n",
      "\\boxed{(a, b) \\text{ such that } a = b \\text{ or } a - b \\text{ is a power of 2}.}\n",
      "\\]\n",
      "\\prod f(a_i)\n",
      "(x, y, z) = (2, 1, 2)\n",
      "\\text{All odd integers}\n",
      "\\text{Penny}\n",
      "f(x) = \\begin{cases}\n",
      "-(1-x)^c + 1 &\\text{ if $x < 1$} \\\\\n",
      "1 &\\text{ if $x = 1$} \\\\\n",
      "k(x-1)^c + 1 &\\text{ if $x > 1$}\n",
      "\\end{cases}, \\text{ for some real constants }c \\geq 0, k > 0\n",
      "\\text{all odd }r\n",
      "\\text{all odd integers }n \\ge 3\n",
      "n \\text{ such that } n+1 \\text{ is prime.}\n",
      "\\text{No such } n\n",
      "n\\text{ not divisible by } 3\n",
      "\\text{all } k\n",
      "$\\text{  no solution }$\n",
      "\\text{Basil}\n",
      "n \\text{ is not a multiple of } 4\n",
      "x=y=z\n",
      "N = 2^a \\cdot 3^b \\text{ for nonnegative integers } a \\text{ and } b\n",
      "d^{3}+e^{3}\\ge a^{3}+b^{3}+c^{3}\n",
      "\\text{Yes}\n",
      "\\text{The answer is that a,b,c are integers summing to }1\n",
      "n \\text{ for all } n > 1\n",
      "\\text{The triangle } ABC \\text{ is right if } h \\leq \\frac{AB}{2}, \\text{ and is isosceles with } AC = BC \\text{ if } h > \\frac{AB}{2}.\n",
      "a_n = \n",
      "\\begin{cases} \n",
      "F_{k+1}^2 & \\text{if } n = 2k, \\\\\n",
      "F_{k+1} \\times F_k & \\text{if } n = 2k-1.\n",
      "\\end{cases}\n",
      "\\text{Yes}\n",
      "(a, b, c) = (238,238,477)\n",
      "\\text{The only } (n, m)\\text{-good set is } \\mathbb{N} \\text{ if and only if } n \\text{ is odd.}\n",
      "N = \\text{the sum of distinct odd powers of }2\n",
      "\\[ (x, y, w, z) = \\boxed{\\left(1, \\frac{1}{2}, -\\frac{1}{2}, \\frac{1}{3}\\right)} \\]\n",
      "{x=m\\pi-\\frac{\\pi}{4}\\ ,\\ m\\in Z}\n",
      "\\text{yes}\n",
      "\\text{It is always possible to choose an adequate } p \\text{ and } m \\text{ to achieve a fair selection.}\n",
      "$如果两张牌不连续（29 和 1 是连续的），则选择每张牌后面的牌；如果两张牌连续，则选择它们后面的两张牌。例如，如果观众选择 4 和 7，则助手选择 5 和 8；如果观众选择 27 和 28，则助手选择 29 和 1。$\n",
      "\\begin{align*}\n",
      "x_1 &= \\frac{(nc_1 + (n-1)c_2 + \\ldots + 2c_{n-1} + c_n)}{(n+1)} \\\\\n",
      "x_2 &= \\frac{(n-1)c_1 + 2((n-1)c_2 + \\ldots + 2c_{n-1} + c_n)}{(n+1)} \\\\\n",
      "x_3 &= \\frac{((n-2)(c_1 + 2c_2) + 3((n-2)c_3 + \\ldots + 2c_{n-1} + c_n))}{(n+1)} \\\\\n",
      "x_4 &= \\frac{((n-3)(c_1 + 2c_2 + 3c_3) + 4((n-3)c_4 + \\ldots + 2c_{n-1} + c_n))}{(n+1)} \\\\\n",
      "& \\vdots \\\\\n",
      "x_{n-1} &= \\frac{(2(c_1 + 2c_2 + \\ldots + (n-2)c_{n-2}) + (n-1)(2c_{n-1} + c_n))}{(n+1)} \\\\\n",
      "x_n &= \\frac{(c_1 + 2c_2 + \\ldots + nc_n)}{(n+1)}\n",
      "\\end{align*}\n",
      "\\text{Kolya wins if Dima starts, and Dima wins if Kolya starts.}\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "655"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from rllm.rewards.math_utils.utils import _normalize, should_allow_eval\n",
    "\n",
    "filtered_data = []\n",
    "count = 0\n",
    "for entry in data:\n",
    "    answer = entry[\"answer\"]\n",
    "    try:\n",
    "        if should_allow_eval(_normalize(answer)):\n",
    "            filtered_data.append(entry)\n",
    "            continue\n",
    "        else:\n",
    "            # print(answer)\n",
    "            count += 1\n",
    "    except Exception:\n",
    "        # print(answer)\n",
    "        count += 1\n",
    "\n",
    "count"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "with open(\"./deepscaler.json\", \"w\") as f:\n",
    "    json.dump(filtered_data, f)"
   ]
  }
 ],
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